0.5 Rational ExpressionsGoal: To Simplify and rationalize the denominator of

rational expressions.

• Rational Expression:• A rational expression is fraction whose

numerator and denominator are polynomials

– For example:

2 5

3

x

x

Rational Functions:

• Let u(x) and v(x) be polynomial functions. The function

Is a rational function. The domain of this rational expression is the set of all real numbers for which v(x) ≠ 0.

( )( )

( )

u xf x

v x

• The key issue with the domain of rational expressions is that we can never divide by zero.– Recall: Dividing by zero is undefined– ex.

4 n

n

Domain = (-∞,0) U (0,∞)

3

2

9

10 21

x x

x x

Domain = (-∞,3) U (3,7) U (7,∞)

• Example:

Whenever we work with rational expressions, we have to make sure we check the domain. We never want to have an answer that results in the function becoming undefined.

Find the domain 2 4

( )2

xf x

x

Find the domain 1( )

1f x

x

• (-∞,2) U (2, ∞)

• (-∞,1) U (1, ∞)

• To simplify rational expressions:– Factor both numerator and denominator– Find domain of denominator– Reduce where possible

• Example:

3

2

9

10 21

x x

x x

3

2

9

10 21

x x

x x

2( 9)

( 7)( 3)

x x

x x

( 3)( 3)

( 7)( 3)

x x x

x x

( 3)

( 7)

x x

x

( 3)( 3)

( 7)( 3)

x x x

x x

(-∞, 3)U(3,7)U(7,∞) is the domain of the denominator

(-∞, 3)U(3,∞) We list this as our domain restriction since we have canceled the factor (x-3) on the graph there is a whole at x = 3

2

-2

-4

-6

-8

-5 5

f x = xx-3 x+3

x-7 x-3

Using your table, see what the value of y is when x = 3, what is different about x = 7?

*This will lead us to removable and non removable discontinuity in ch. 1

• Example: Simplify

3 2

2

3 21 18

3 3

x x x

x x

23 ( 7 6)

3 ( 1)

x x x

x x

3 ( 6)( 1)

3 ( 1)

x x x

x x

6x

(-∞, 0)U(0,1)U(1,∞) is the domain of the denominator

x Є (-∞, 0)U(0,1)U(1,∞)

• To Multiply/Divide rational expressions:– Factor both numerator and denominator– Find domain of denominator for both expressions (also

denominator of reciprocal when you divide)– Reduce where possible

• Example:

2

2 3

3 5 6

x x

x x x

( 2) ( 3)

( 3) ( 2)( 3)

x x

x x x

( 2) ( 3)

( 3) ( 2)( 3)

x x

x x x

1

( 3)x

x Є (-∞, -3) U(-3,2)U(2,3)U(3,∞)

x Є (-∞,2)U(2,3)U(3,∞)

8 ( 1)

3

x x

x Є (-∞, -1) U(-1,0)U(0,1)U(1,∞)

2

2 3

2 12 3 18

4 4

x x x

x x x x

EX.

2

2 ( 6) 3( 6)

( 1) 4 ( 1)

x x x

x x x x

2 ( 6) 3( 6)

( 1) 4 ( 1)( 1)

x x x

x x x x x

2 ( 6) 4 ( 1)( 1)

( 1) 3( 6)

x x x x x

x x x

x Є (-∞, -6) U(-6, -1) U(-1,0)U(0,1)U(1,∞)

• To Add/Subtract rational expressions:– Factor denominators– Find domain of denominator for both expressions– Reduce where possible– Find Common Denominator

• Smiley Face• Other methods: What’s missing, etc.

• Example:

2

2

4 2

x

x x

2

2

4 2

x

x x

• Example:

2

1 1

3( 2 ) 3x x x

• Rationalizing Techniques:– Use the “conjugate”

• Example: 1

2x x

• Rationalizing The Numerator:– Example

15 3

12